ROTATIONAL MATCHING PROBLEMS
Bibliographic record
Abstract
This paper addresses the issue of obtaining the optimal rotation to match two functions on the sphere by minimizing the squared error norm and the Kullback–Leibler information criteria. In addition, the accuracy in terms of the band-limited approximations in both cases are also discussed. Algorithms for fast and accurate rotational matching play a significant role in many fields ranging from computational biology to spacecraft attitude estimation. In electron microscopy, peaks in the so-called "rotation function" determine correlations in orientation between density maps of macromolecular structures when the correspondence between the coordinates of the structures is not known. In X-ray crystallography, the rotational matching of Patterson functions in Fourier space is an important step in the determination of protein structures. In spacecraft attitude estimation, a star tracker compares observed patterns of stars with rotated versions of a template that is stored in its memory. Many algorithms for computing and sampling the rotation function have been proposed over the years. These methods usually expand the rotation function in a bandlimited Fourier series on the rotation group. In some contexts the highest peak of this function is interpreted as the optimal rotation of one structure into the other, and in other contexts multiple peaks describe symmetries in the functions being compared. Prior works on rotational matching seek to maximize the correlation between two functions on the sphere. We also consider the use of the Kullback–Leibler information criteria. A gradient descent algorithm is proposed for obtaining the optimal rotation, and a measure is defined to compare the convergence of this procedure applied to the maximal correlation and Kullback–Leibler information criteria.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.018 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.005 | 0.003 |
| Insufficient payload (model declined to judge) | 0.020 | 0.005 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".